Yes, it's possible to get a teaching job in China without a degree if you hold a certificate. Is it legal to do so? Not really, but that doesn't always matter. Most Teaching English as a Second or Foreign Language programs require that you have acquired at least a BA; however, many such programs have sprung up - online and elsewhere, that don't have that prerequisite. Personally, I wouldn't recommend any of them, as they're not worth the paper they are printed on. I've even heard that in places such as Bangkok you can buy a TEFL Certificate that looks authentic without having to attend one class. Of course, people have been known to do that with BA degrees or MA's as well. Hey, good idea, I think I'll buy my PHD online.

But back to your original question. Yes, you can get a job, but not in a public school or university where your documents are generally scrutinized pretty carefully.

I'm not sure about Shanghai. By my way of thinking, Shanghai is a whole separate country - part of China by name only. In fact, many of my Chinese friends, when the subject of Shanghai comes up, tell me that Shanghai citizens generally hold themselves above the rest of the country - they look down on anyone not from Shanghai as if they are mere country bumpkins unworthy of their respect. But in regards to teaching privately there, I'm sure it's doable - legal or not. If it's not legal, a few well-placed remimbi will pave the way. I imagine though that it's highly competitive. A lot of ex-pats earn extra cash that way and there are many, many private EFL schools where many well-qualified teachers are teaching.
Having said all the above though, my advice - if you really want to experience China - is to stay away from Shanghai unless you just want to use it as a base from which you can travel to the "real" China.